On the maximum ABC index of graphs without pendent vertices

Let G be a simple graph. The atom–bond connectivity index (ABC) of G is defined as A B C ( G ) = ∑ u v ∈ E ( G ) d ( u ) + d ( v ) − 2 d ( u ) d ( v ) , where d(v) denotes the degree of vertex v of G. We characterize the graphs with n vertices, minimum vertex degree ≥ 2, and m edges for m = 2 n − 4 and m = 2 n − 3 , that have maximum ABC index.